The Helix Tour

Welcome to the tour. Throughout this tour the formulas will be shown as we demonstrate our exact procedures of this project. If you would like to see only the formulas please click on the 'Formulas button on your left.. Thank you. On We Go!

Note: To properly use the vector functions you must be aware of possible calculations we apply to these vectors. A few examples would be cross product, dot product, magnitude, differentiation... If you don't know how to perform operations on vectors, please click on the 'Help with vectors' link below. Thank you.

As you may (or may not ) know, a helix is based on a type of function known as a Vector-Valued Function. The definition as given in the book Calculus is:

Definition of a Vector-Valued Function
A function of the form
  r(t) = f(t)i + g(t)j
  r(t) = f(t)i + g(t)j + h(t)k
is a vector-valued function, where the component functions f,g, and h, are real-valued fucntions of the parameter t. Vector-valued functions are sometimes denoted as
r(t) = <f(t),g(t)> or r(t) = <f(t),g(t),h(t)>.
Help with vectors

Because all functions are based on the value 't' , we can use this vector-valued function to represent motion or a trace along a curve.When we graph these 'traces' along the curve we get what is defined to be a Helix! In the case of the parking garage, we got the formula of our position function (in inches):

R(t) = 423 Cos(t)i + 423 Sin(t)j + 36tk

After finding the function of the garage's helix we then found it's velocity and acceleration functions. We did this by taking the first derivative of our position function R(t) to get velocity and the second derivative of R(t) to get acceleration.

v(t) = R'(t) = -423 Sin(t)i + 423 Cos(t)j + 36k
a(t) = R''(t) = -423 Cos(t)i - 423 Sin(t)j

Once we have found our velocity vecotr and acceleration vector, we can then proceed to find our TNB coordinate system. What is a TNB coordinate system you ask? Well, since you asked...A TNB coordinate. system is a point on the helix where we have a 'unitized' vector coordinate system. Basically, this allows us to 'ride' the helix from the point's point of view. The 'T' in TNB represents the unit tangent vector at the specified point. This shows us in what direction the tangent to the point is directed ( remember it is a unit vector so it is only 1 unit long). The 'N' in TNB is the principal unit normal vector at the specified point. The principal unit normal shows us in which direction the curve is turning. The 'B' in the TNB represents the unit binormal vector at the specified point. This binormal vector is the blah blah blah. To obtain your TNB system you must use the formulas below.

Note: || R'(t) || and || T '(t) || both represent magnitudes and the T x N represents a cross product. If you don't know what magnitudes or cross products are plese see the note at the top of the page ( marked in red). Thank you.

Our TNB System looks like this:

T(t) = R'(t) / || R'(t)|| = -.9964 Sin(t)i + .9964 Cos(t)j + .0848k
N(t) = -Cos(t)i - Sin(t)j + 0k
B(t) = .0848 Sin(t)i - .0848Cos(t)j + .9964k

To Be Continued...